The Logitech wheel for PC works with Gran Turismo 3 so you don't need to buy the PS2 version. Some games let you use a mouse and keyboard, so you don't need to purchase an official set. THPS3 supports USB cable modems, so you can use those.

No, if you have a cable modem or DSL router, you use the one you already have. THPS3 can use a USB ethernet adaptor in conjunction with this. It also can use a USB modem for direct dial. But it does not use a "USB Cable Modem".

I meant that you can use the USB ports to connect to a cable modem instead of having to buy the PS2 adapter. I'm not saying you could go directly to the modem using the USB, but you can get to it through them (however thats done, I'm not exactly trying to be specific).
What game are you using the wheel on? GT3 works with the Logitech wheel (its the GT3 PS2 wheel without the GT3 logo, AFAIK) but thats it.